On a question of Dusa McDuff

28 Pages·0.29 MB·

by  Felix Schlenk

ON A QUESTION OF DUSA MCDUFF
FELIX SCHLENK
Abstract. Consider the 2n-dimensional closed ball B of radius
1 in the 2n-dimensional symplectic cylinder Z = D× 2n−2 over
2 R
the closed disc D of radius 1. We construct for each ǫ > 0 a
0
Hamiltonian deformation ϕ of B in Z of energy less than ǫ such
0
2 that the area of each intersection of ϕ(B) with the disc D×{x},
x∈ 2n−2, is less than ǫ.
n R
a
J

1

1. Introduction
]
G
We endow Euclidean space 2n with the standard symplectic form
S
R
h. n
t ω = dx ∧dy .
a 0 i i
m Xi=1
[ A C∞-smooth embedding ϕ of an open subset U of 2n into 2n is

1 called symplectic if ϕ∗ω = ω . An embedding of an aRrbitrary Rsubset
0 0
v S of 2n into another subset S′ of 2n is called symplectic if it extends
6 R R
0 to a symplectic embedding of a neighbourhood of S into 2n. We
R
0 denote by B2n(πr2) the closed 2n-dimensional ball of radius r and by
1
Z2n(π) the closed 2n-dimensional symplectic cylinder
0
2 Z2n(π) = B2(π)× 2n−2.
0
R
/
h Gromov’s celebrated Nonsqueezing Theorem [3] states that there does
t
a not exist a symplectic embedding of the ball B2n(a) into the cylinder
m
Z2n(π) if a > π. So fix a ∈ ]0,π]. We recall that the simply con-
:
v nected hull T of a subset T of 2 is the union of its closure T and the
Xi bounded components of 2\T.RWe denote by µ the Lebesgue measure
b R
r on 2, and we abbreviate µˆ(T) = µ T . It is well-known that the
a R
Nonsqueezing Theorem is equivalent t(cid:0)o e(cid:1)ach of the identities
b
a = inf µ p ϕ(B2n(a)) ,
ϕ
(cid:0) (cid:0) (cid:1)(cid:1)
a = inf µˆ p ϕ(B2n(a)) ,
ϕ
(cid:0) (cid:0) (cid:1)(cid:1)
Date: February 8, 2008.
2000 Mathematics Subject Classification. Primary 53D35, Secondary 57R40.
1
2 FELIX SCHLENK
where ϕ varies over all symplectic embeddings of B2n(a) into Z2n(π)
and where p: Z2n(π) → B2(π) is the projection, see [1] and [9, Corol-
lary B.10]. Following [7, Section 3] we consider sections of the image
ϕ(B2n(a)) instead of its projection, and define
σ(a) = inf sup µ p ϕ(B2n(a))∩D ,
x
ϕ x
(cid:0) (cid:0) (cid:1)(cid:1)
σˆ(a) = inf sup µˆ p ϕ(B2n(a))∩D ,
x
ϕ x
(cid:0) (cid:0) (cid:1)(cid:1)
where ϕ again varies over all symplectic embeddings of B2n(a) into
Z2n(π), and where D ⊂ Z2n(π) denotes the disc D = B2(π)×{x},
x x
x ∈ 2n−2. Clearly,
R
σ(a) ≤ σˆ(a) ≤ a.
It is also well-known that the Nonsqueezing Theorem is equivalent to
the identity
(1.1) σˆ(π) = π.
Indeed, the Nonsqueezing Theorem implies that for every symplectic
embedding ϕ of B2n(π) into Z2n(π) there exists x ∈ 2n−2 such that
R
ϕ(B2n(π))∩D containstheunitcircleS1×{x},see[6,Lemma1.2]. On
x
her search for symplectic rigidity phenomena beyond the Nonsqueezing
Theorem, D. McDuff therefore asked for lower bounds of the function
σ(a) and whether σ(a) → π as a → π. It was known to L. Polterovich
that σ(a)/a → 0 as a → 0, see again [7]. We shall prove
1.1. Theorem.
(i) σ(a) = 0 for all a ∈ ]0,π].
(ii) σˆ(a) = 0 for all a ∈ ]0,π[.
The symplectic embeddings in the definition of σ(a) and σˆ(a) were not
further specified. Following a suggestion of L. Polterovich, we next ask
whether the vanishing phenomenon described by Theorem 1.1 persists
if we restrict ourselves to symplectic embeddings which are close to
the identity mapping in a symplectically relevant sense. We denote by
H (2n) the set of smooth functions H: 2n → whose support is a
c
R R
compact subset of Z2n(π). For H ∈ H (2n) we define the Hamiltonian
c
vector field X through the identities
H
ω (X (z),·) = dH(z), z ∈ 2n,
0 H
R
and denote by φ the time-1-map of the flow generated by X . More-
H H
over, we abbreviate
(1.2) kHk = sup H(z)− inf H(z).
z∈ 2n z∈ 2n
R R
3
For each a ∈ ]0,π] we define
σ (a) = inf sup µ p φ (B2n(a))∩D +kHk ,
H H x
H (cid:26) x (cid:27)
(cid:0) (cid:0) (cid:1)(cid:1)
σˆ (a) = inf sup µˆ p φ (B2n(a))∩D +kHk ,
H H x
H (cid:26) x (cid:27)
(cid:0) (cid:0) (cid:1)(cid:1)
where H varies over H (2n). Clearly, σ(a) ≤ σ (a) and σˆ(a) ≤ σˆ (a).
c H H
In particular, σˆ (π) = π.
H
1.2. Theorem.
(i) σ (a) = 0 for all a ∈ ]0,π].
H
(ii) σˆ (a) = 0 for all a ∈ ]0,π[.
H
In order to see Theorem 1.2 in its right perspective we abbreviate
Ham Z2n(π) = {φ | H ∈ H (2n)}
c H c
and define the energy(cid:0)E(φ) of(cid:1)φ ∈ Ham (Z2n(π)) by
c
E(φ) = inf{kHk | φ = φ for some H ∈ H (2n)}.
H c
In the framework of Hofer geometry the energy of a Hamiltonian dif-
feomorphism is its distance from the identity mapping, see [5, 6, 8].
Notice that
σ (a) = inf sup µ p φ(B2n(a))∩D +E(φ) ,
H x
φ (cid:26) x (cid:27)
(cid:0) (cid:0) (cid:1)(cid:1)
σˆ (a) = inf sup µˆ p φ(B2n(a))∩D +E(φ) ,
H x
φ (cid:26) x (cid:27)
(cid:0) (cid:0) (cid:1)(cid:1)
where φ varies over Ham (Z2n(π)). Theorem 1.2 therefore says that
c
the vanishing phenomenon described by Theorem 1.1 persists if we
restrictourselvestoHamiltoniandiffeomorphismofZ2n(π)whoseHofer
distance to the identity mapping is arbitrarily small.
2. Results
We start with stating a generalization of Theorem 1.1. We denote by µ
the outer Lebesgue measure on 2 and by µˆ(T) = µ T the Lebesgue
R
measure of the simply connected hull of the subset T(cid:0) o(cid:1)f 2. For each
b R
subset S of the cylinder Z2n(π) we define
σ(S) = inf sup µ(p(ϕ(S)∩D )),
x
ϕ x
σˆ(S) = inf sup µˆ(p(ϕ(S)∩D )),
x
ϕ x
where ϕ varies over all symplectic embeddings of S into Z2n(π). We
abbreviate the closed cylinder Z2n(a) = B2(a)× 2n.
R
4 FELIX SCHLENK
2.1. Theorem. Consider a subset S of Z2n(π).
(i) σ(S) = 0.
(ii) σˆ(S) = 0 if S ⊂ Z2n(a) for some a < π.
In view of the identity (1.1) we have σˆ(S) = π whenever S contains
the ball B2n(π).
2.2. Question. Is it true that σˆ(IntB2n(π)) = π?
Aslightlyweaker versionofTheorem2.1hasbeenprovedin[9]byusing
asymplectic foldingmethod. Themethodusedhereismoreelementary
and can also be used to prove a generalization of Theorem 1.2. We
denotebyH(2n)thesetofsmoothandboundedfunctionsH: 2n →
R R
whose support is contained in Z2n(π) and whose Hamiltonian vector
field X generates a flow on 2n. The time-1-map of this flow is then
H
R
againdenotedbyφ . Usingthenotation(1.2)wedefineforeachsubset
H
S of Z2n(π),
σ (S) = inf sup µ(p(φ (S)∩D ))+kHk ,
H H x
H (cid:26) x (cid:27)
σˆ (S) = inf sup µˆ(p(φ (S)∩D ))+kHk ,
H H x
H (cid:26) x (cid:27)
where H varies over H (2n) if S is bounded and over H(2n) if S is
c
unbounded. In order to state the main result of this note we need yet
another definition.
2.3. Definition. A subset S of Z2n(π) is partially bounded if at least
one of the coordinate functions x ,...,x ,y ,...,y is bounded on S.
2 n 2 n
2.4. Theorem. Consider a partially bounded subset S of Z2n(π).
(i) σ (S) = 0.
H
(ii) σˆ (S) = 0 if S ⊂ Z2n(a) for some a < π.
H
Of course,
σ Z2n(π) = σˆ Z2n(π) = σ IntZ2n(π) = σˆ IntZ2n(π) = π.
H H H H
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
2.5. Question. Is it true that σ (Z2n(a)) = σˆ (Z2n(a)) = a for all
H H
a ∈]0,π]?
Theorem 2.1 and Theorem 2.4 are proved in the next two sections.
In Section 5 we shall reformulate these theorems in the language of
symplectic capacities.
5
Acknowledgements. I cordially thank David Hermann and Leonid
Polterovich for a number of helpful discussions. I wish to thank Tel
Aviv University for its hospitality and the Swiss National Foundation
for its generous support.
3. Proof of Theorem 2.1
The mainingredient inthe proof ofTheorem 2.1 is aspecial embedding
result in dimension 4. We shall use coordinates z = (u,v,x,y) on
( 4,du∧dv+dx∧dy). We denote by E ⊂ 4 the affine plane
(x,y)
R R
E = 2 ×{(x,y)},
(x,y)
R
and given any subset S of 4 we abbreviate
R
µ S ∩E = µ p S ∩E , µˆ S ∩E = µˆ p S ∩E .
(x,y) (x,y) (x,y) (x,y)
(cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
Fix an integer k ≥ 2. We set
π ǫ
ǫ = , δ = ,
k 4k
and we define closed rectangles P, P′ and Q in 2(u,v) by
R
P = [0,π]×[0,1],
P′ = [δ,π −δ]×[δ,1−δ],
Q = [3δ,π −3δ]×[3δ,1−3δ].
We abbreviate the support of a map ϕ: 4 → 4 by
R R
supp ϕ = {z ∈ 4 | ϕ(z) 6= z}.
R
3.1. Proposition. There exists a symplectomorphism ϕ of 4 such
that supp ϕ ⊂ P′ × 2 and such that for each (x,y) ∈ 2, R
R R
′
(3.1) µ ϕ(P × ×[0,1])∩E ≤ 2ǫ,
(x,y)
R
(cid:0) (cid:1)
(3.2) µˆ ϕ(Q× ×[0,1])∩E ≤ 2ǫ.
(x,y)
R
(cid:0) (cid:1)
Proof. We define closed rectangles R, R′ and R′′ in 2(u,v) by
R
R = [0,ǫ]×[0,1],
R′ = [δ,ǫ−δ]×[δ,1−δ],
R′′ = [2δ,ǫ−2δ]×[2δ,1−2δ],
and we define closed rectangular annuli A and A′ in 2(u,v) by
R
A = R\R′, A′ = R′ \R′′.
Then R = A∪A′ ∪R′′, cf. Figure 1.
6 FELIX SCHLENK
v
1
δ
u
δ ǫ
Figure 1. The decomposition R = A∪A′ ∪R′′.
We choose smooth cut off functions f ,f : → [0,1] such that
1 2
R
0, t ∈/ [δ,ǫ−δ],
f (t) =
1 (cid:26) 1, t ∈ [2δ,ǫ−2δ],
0, t ∈/ [δ,1−δ],
f (t) =
2 (cid:26) 1, t ∈ [2δ,1−2δ],
and we define the smooth function H: 4 → by
R R
H(u,v,x,y) = −f (u)f (v)(1+ǫ)x.
1 2
The Hamiltonian vector field X of H is given by
H
−f (u)f′(v)x
1 2
 f′(u)f (v)x 
(3.3) X (u,v,x,y) = (1+ǫ) 1 2 .
H
0
 
 
 f (u)f (v) 
1 2
 
The time-1-map φ has the following properties.
H
(P1) supp φ ⊂ R′ × 2,
H
R
(P2) φ fixes A× 2,
H
(P3) φ embeds AR′ × 2 into A′ × 2,
H
(P4) φ translates R′′R× 2 by (1+Rǫ)1 .
H y
R
where we abbreviated 1 = (0,0,0,1).
y
For each subset T of 2(u,v) and each i ∈ {1,...,k} we define the
R
translate T of T by
i
T = {(u+(i−1)ǫ,v) | (u,v) ∈ T}.
i
7
With this notation we have
k k
′ ′′
P = R = A ∪A ∪R ,
i i i i
i[=1 i[=1
v
1
δ
u
δ ǫ π
k k
Figure 2. The decomposition P = R = A ∪
i=1 i i=1 i
A′ ∪R′′ for k = 4. S S
i i
cf. Figure 2. Abbreviate H (u,v,x,y) = iH(u − (i − 1)ǫ,v,x,y). We
i
define the smooth function H: 4 → by
R R
k
e
H(z) = H (z)
i
Xi=1
e
and we define the symplectomorphism ϕ of 4 by ϕ = φ . In view of
R H
the identity (3.3) we see that ϕ is of the form
e
′ ′ ′
(3.4) ϕ(u,v,x,y) = (u,v ,x,y ),
and in view of the Properties (P1)–(P4) we find
P1 supp ϕ ⊂ P′ × 2,
R
(cid:0)P2(cid:1) ϕ fixes k A × 2,
f i=1 i R
(cid:0)P3(cid:1) ϕ embeSds A′ × 2 into A′ × 2, i = 1,...,k,
f i i
R R
(cid:0)P4(cid:1) ϕ translates R′′ × 2 by i(1+ǫ)1 , i = 1,...,k.
f i y
R
(cid:0) (cid:1)
Vferification of the estimates (3.1) and (3.2)
Fix (x,y) ∈ 2. We abbreviate
R
P′ = p ϕ(P′ × ×[0,1])∩E ,
(x,y)
R
Q = p(cid:0)ϕ(Q× ×[0,1])∩E (cid:1).
(x,y)
R
(cid:0) (cid:1)
3.2. Lemma. We have µ(P′) ≤ 2ǫ.
8 FELIX SCHLENK
y
5+4ǫ
4+3ǫ
3+2ǫ
2+ǫ
1+ǫ
1
u
2δ ǫ π
Figure 3. The intersection of ϕ(P × ×[0,1]) with a
R
plane {(u,v,x,y) | v,x constant} for v ∈ [2δ,1−2δ].
Proof. Using the definitions ǫ = π and δ = ǫ we estimate
k 4k
ǫ
′
(3.5) µ(A ∪A ) = ǫ−(ǫ−4δ)(1−4δ) ≤ , i = 1,...,k.
i i k
Case A: y ∈ [i∗(1+ǫ),i∗(1+ǫ)+1]. AccordingtoProperties P2 – P4
we have P′ ∩R′′ = ∅ if i 6= i∗, and so (cid:0) (cid:1) (cid:0) (cid:1)
i f f
k
′ ′
P ⊂ Ri∗ ∪ Ai ∪Ai.
i[=1
Together with the estimate (3.5) we therefore find
ǫ
′
(3.6) µ P ≤ ǫ+k = 2ǫ.
k
(cid:0) (cid:1)
k
Case B: y ∈/ [i(1 + ǫ),i(1 + ǫ) + 1]. According to Properties
i=1
P2 – P4 we hSave P′ ∩R′′ = ∅ for all i, and so
i
(cid:0) (cid:1) (cid:0) (cid:1)
f f k
′ ′
P ⊂ A ∪A .
i i
i[=1
9
Therefore,
′
(3.7) µ(P ) ≤ ǫ.
The estimates (3.6) and (3.7) yield that µ(P′) ≤ 2ǫ. 2
3.3. Lemma. We have µˆ(Q) ≤ 2ǫ.
Proof. In view of the special form (3.4) of the map ϕ we have
Q = p ϕ(Q×{x}×[0,1])∩E .
(x,y)
(cid:0) (cid:1)
For i = 1,...,k we abbreviate the intersections
′ ′ ′′ ′′
(3.8) A = Q∩A , A = Q∩A , R = Q∩R .
i i i i i i
v
1
3δ
u
3δ π
Figure 4. The subsets A , A′ and R′′ of Q, i = 1,...,4.
i i i
Each of the sets A and A′ consists of one closed rectangle if i ∈ {1,k}
i i
and of two closed rectangles if i ∈ {2,...,k − 1}, cf. Figure 4. The
crucial observation in the proof is that for each i the simply connected
hull of the part
′
p ϕ(A ×{x}×[0,1])∩E
i (x,y)
of Q is a simply con(cid:0)nected subset of A′. Indeed(cid:1), according to prop-
i
erty P3 the closed and simply connected set ϕ(A′ ×{x}×[0,1])
i
is con(cid:0)tain(cid:1)ed in A′ × {x} × , and so the simply connected hull of
ϕ(A′ ×f{x}×[0,1i])∩E isRasimplyconnectedsubsetofA′×{(x,y)}.
i (x,y) i
We abbreviate by Q the simply connected hull of Q.
Case A: y ∈ [0,1]. Abccording to Properties P2 – P4 we have Q∩
A = A and Q∩R′′ = ∅ for all i. In view of t(cid:0)he a(cid:1)bo(cid:0)ve o(cid:1)bservation we
i i i f f
conclude that
k
′
Q ⊂ A ∪A .
i i
i[=1
b
10 FELIX SCHLENK
Together with the estimate (3.5) we therefore find
ǫ
(3.9) µ Q ≤ k = ǫ.
k
(cid:0) (cid:1)
b
Case B: y ∈ [i∗(1+ǫ),i∗(1+ǫ)+1]. According to Properties P2 –
P4 we have Q∩A = ∅ for all i and Q∩R′′ = ∅ if i 6= i∗. In v(cid:0)iew(cid:1)of
i i f
(cid:0)the(cid:1)above observation we conclude that
f
k
′
Q ⊂ Ri∗ ∪ Ai.
i[=1
b
Therefore,
(3.10) µ Q ≤ ǫ+ǫ = 2ǫ.
(cid:0) (cid:1)
k b
Case C: y ∈/ [0,1]∪ [i(1+ǫ),i(1+ǫ)+1]. According to Properties
i=1
P2 – P4 we have QS∩A = Q∩R′′ = ∅ for all i. In view of the above
i i
(cid:0)obse(cid:1)rv(cid:0)atio(cid:1)n we conclude that
f f
k
′
Q ⊂ A .
i
i[=1
b
Therefore,
(3.11) µ Q ≤ ǫ.
(cid:0) (cid:1)
The estimates (3.9), (3.10) and (b3.11) yield that µˆ(Q) = µ Q ≤ 2ǫ.
2
This completes the proof of Lemma 3.3. (cid:0) (cid:1)
b
In view of Lemmata 3.2 and 3.3 the estimates (3.1) and (3.2) hold
2
true. The proof of Proposition 3.1 is thus complete.
End of the proof of Theorem 2.1(i)
Fix k ≥ 2 and set ǫ = π. We choose a symplectomorphism α of
k
2(u,v) such that P′ ⊂ α(B2(π)). We refer to [9, Lemma 2.5] for an
R
explicit construction. Chooseanorientationpreserving diffeomorphism
f: → ]0,1[ and denote by f′ its derivative. Then the map
R
x
β: 2 → ×]0,1[, (x,y) 7→ , f(y)
R R (cid:18)f′(y) (cid:19)
isasymplectomorphism. WedefinethesymplecticembeddingΦ: 2n ֒→
R
2n by
R
Φ = (α−1 ×id)◦ϕ◦(α×β) ×id
2n−4
(cid:0) (cid:1)